all (Result) (Case)to (Result), (Case)and (Result) t,o (R.ule)

the Japan Association for Philosophy of Science

NII-Electronic Library Service

the JapanAssociation forPhUosophy ofScience

Anllals ofthe Japan Association forPhilosoph,y ofScjence Vol.17 (2009) 1'Lll 1

A Mathematical Model of Deductive

Non-Deduetive Inferences

A,fakc}to KIKuCIII"

and

Abstract

Induction and abduction are well known llon-deduetive inferenccs. We shall

propose a vie", Lhat design is also another foriiL of non-deductive infereiice, and

give a i-athetnatical rnodel of cleductive a,nd non-deductive inferences based on

Barwise and Seligman;s mat,heinat･ical Lheory of informa,tien fiow. Iii our rnoclcl,

inferences are classified into three ca,tcgories, and we cari sho", that deducLioii a,Tid

abduction arc in t,he same category, a]thoiigh inducLlon is different. Furthermorc,

we sha.Il show also that non-deductive infey'eiices are interpret･ablc mutuaily, and

invcstigate also inatheina,tic:al properties of the model. In particular, we shall

prove a genoralized ver$ion of the Abstract Completeness "1'heorem

by Barwise

and Seligman.

Key words: Deducticm, Induction, Abduction, Channcl Thaory

1. Introduction

Ill the lat,e 19tli ceTitury, Pei]'ce classified inferences in two categories, ded'u･c-

tive (or a･n.a,lytic) inferences and synthetic inferences, and divided synthetic infereiices

into induetion alld h!ypoLhesis [111, where hypot･hesis was ln,ter rei.tarned abduction or

retroditction. The following i$ Peirce's example in [11].

Suppose I enter a, ruoin and there find a nuinber of bags, conta,ining different

kinds of beaiis, On the t･able t,here is a, handful ef white beansl and, after

some searching, I find one of the bags cont･ains w･hite beans only. 1 at once

infer as a probability, or as a fair guess, tliat this handful was t･aken out of

that bag. T'his sort of illfereiice is called ma･king a,n hypothesis.

T'his exarnple consists of t･hree a6sertions, which are called (Rule) `:all

the bea-s

fi'om t,his bag are white;', (Case) `tthese

beans are firom this bag", and (Result) :Lthese

beans are white;'. Then, Peirce defined dcduct･ion a,s an inforence frem (Rule) and

(Case) to (Result), induction as an inference from (Case) and (Result) t,o (R.ule), and

' Department of Compul,er Scicnce and S.vstems Engineering Kobe IJniversity

E-mail: mkikuchi(l/kobe-u.ac,jp

-- 1-



the JapanAssociation forPhUosophy of Science

2 pt'lakoto I<.1.K u/ c:HI Vol. 17

hyx)ot.hesis as an inrerellce frolli (Rule) and (Resul't) to (Case). "t'e can forma]ize these inferellces within the fra,mework of first-order logic, but

it is not easy t･o distinguish induction and a,hcluction. Flach and Kakas pointed out in

[71 t･hat the definition ef abcluction by Kakas et, al, [8] is, superficially, equivalellt t･o

the definit,ion of induction by )'luggelton a,nd De Rardt l' lO]: ft)r a background t･heory

V' and obserir･ation (], both the abductive explanat･ion and the inductive hy. pothesisare the set, of sentences H such t･hat TJ U' F G and TU L( is consistent. N･Ve have te

gi've a condit,],oll for el's being all a,l)ductive explanat･ion or an illductive hypothesis in

order t･o distinguish abductioii and induetion. Peiree ",ret,e in [11] that `LThe

analogy

of h,vpothcsis with induction is so strollg that some logicians have confusecl t,hen)",

and, recently, van Benthem wrot,e tha,t `'Indeed,

it is not ea,sy t,o give a ci'ystal-clear

definition of theni, either independent/1.v ot' in their inter-relationslLip. (Of course, this

is not eas,y for [deduct･ior]:either'.) 114]':

Peirce ac]dresscd that "ioggc

!s the ar( ef reasoning", and '±･reasoni,ng

ts the pro-ccss by which we aLtain a belief "ihich ",e regard as t･he resu].t Qi' previous kno",ledge"

[121. Tihen, 1'te continued Lhat 'infErence is `[the c,onscioiEs allcl cont,rolLed adoptien of

a belief as a co]'iseqiience of other knowledgei:. "J'e reTnark t,hat de,sign is a, kind of

non-deductive i]iference in this sense. Consider the fo1]owing example.

Suppuse that we waiit to have white beans, If we know that all the beans

in this bag are "rhit･e, "re illfer t,ha,t "J･hat･ "re need to do is to piek out soiiie

beans frorn that bag.

./Vthough this example is too simple, "･e can say t･ha{, this is an cxainple ol' a sort

of design, sillce, in essence., design is an activit.v of obLaining a,rtifacts (bea'rL･s in the

exaTr]ple) which satisf.v a g'iven specifica,tioii (bcz'rtg whitc ir) t･he example). Ybshikaw･tL

pointcd out in [16] t,hat abduction is the key notion for unde:standing design activ-

it.y. Kikuchi discussed a sort oi' dualit,y of analysis and design in [91 in the context

of Channe] Theory, a formal t,heor>, of int'ori'natioi] H()"r 1)y Bar",ise alld Seligintxn

l31, . C]arifyiTig the relai,ioTiship bctwee- a,bduction aTLd clesjgn is a new problem. The

purpose of t･his paper is to propose a mathematical model ef non-deductive inferences

within the framework of Channel Theory. ]n this model, induction, abduct･ion, and

design are three different kinds of noll-deductive inferellc:es. "･"e s'hali show also some

of t･heir inat･hematica! propert･ies. In partici.Llar, X,V'e shall show that these three kinds

of inferences are int.erpretable mut,uaily, and t.his is c)ne of t･he reasons. v･rhy it is not･

eas.v to distinguisli thein.

2. InformationContexts

CJhannel [I]'heory is a ina,the.matica,I theoT'y of informatioii fiow cleveloped hy Bar-

w･ise and Seligmaii 'm

l3j', "rhj.ch is based oii thephilc)sophical diseiission about knourl-

2-



the JapanAssociation for PhUosophy of Science

A pt'rat.heintt,tical )"'[odel of Deductlve and Nori-DeducLive lii'ferences 3

edge a,nd informatioii flow by･ Dretske [5], Aii inforinat･ion coiitext defiiied by Barwise

in [1], which irs called a local logic ill i3], is one of the ke}r notions in Challnel Theory, ,and three kinds of information abuiit two set,s, tL set of tokens ai]d a, set of t･y. pes, are

discussed xNJ･ith inforination contexts. Iii this section, "re shall revie"r basic notioiis

about information context･s.

Definition 2.1. A cla,ssijicat･ion is a t,riple A = (SA, EA. t=A> where F=A is a billary

relation between twe sets SA and E]A. EIemenrs of SA and I!]A are called tokfms and

types of A.

XSJ'e shall on'Lit, t,he subscr'ipt A if it is clear fi'om t,he cont,ext. X･V'e can find a

source ef t,he conc;ept ot' classilications in Situatioi] Theor},, a theory fbr semantics

oi' natura] languag.e invent,ed b.v, Ba･rwise and Perry [2]i tokens and t.ypes are ca}led

si,tua.ti,ons aiid ･irofo･n.,

a,nd =-- is the s'tt･pporL r'elation between t,hcnn.

Let A = <SA,£ A,kA> be a classification a]/.td a c SA. The tsLp,e set t･.v, pA({z)and Lhe sta,Le clescri,f)tion st,ateA(a) oi' a is def[ined by typA(a･) = {a C >]A : a t ctz1},

and st･ateA(a･) = <t",'pACa), EA X t,ypA(ct)>. "J'e shall omit the sub$eript A if it is clear

froi.[i t,he c:oiit,ext,.

Definition 2.2. A sequent on a set, 1; is a pair <I",A> of subset,s ef E], and the set

of sequents on £ is denoted by Seq(E). A thc,or'y on Ev･ is a pair T = <E]･･t･, F'T> ol'

a set >)/t and a subset F'i･ of Seq(Zv ). Elements ot' E･T and e･i- are called types and

constraints of 1i. X･XJ'e write r FT A when <I', A> is a constJ'a･int of T.

I- Fni- A n]eaiis F inLp]ies A: if every eleineiit iTi r is t･rue, t]'ieu sorrie ele-

mellL of A is true. "J'e sas.r <I',A> is T-cons'gstent if r' VT A and T-t'nc:o'r-sistent if

1" FT A. T i,s consi.stent if there is a ILconsisteiit･ sequent, and inconsisteTLt ot･her"rise.

{a, ,3} FT b･, b'} is abbreiiated br,r dv, ,S F"i･ 1･,6, etc.

We sa.},･ <F,A> is a paT'tition if FUA = £ a,iid T r)

A == e,, and <r',A'> is an

e.xteus･ion of <l',A> and write <I-,A> <.. <F',A'> when I" C r' and A {! A', VLr'e say a

token a 7'ca,lizes a sequent (I', A> c)n >:A if <I', A> f{.l. stateA(a,), ancl "re say a scLtis.fies

<T, A) otherwise. That is, a r ¢ alizes <r, A> when a t :,' for al] tv･ c r' a,nd a Vi 5 I'or all

i c A, and a, satisfies <T,A> when a t :F' for all 1･ E r implies a, I.=' ti fot' some ti E A.

Definition 2.3. 1' is said tu be 'r'cg'ula,7'

if and oT]ly if 7- satisfies the [ollowing condi-

tiOIIS:

Identity: cv }-r cv l'or a,ny a E =.

Weakening: 'Let,

r,A.1",A' g £ . If r Lv･

A and <T,A> ( <1-',A'> then

r"-v･ A'.

Global Cut: Let P,A,Z' {. >l. If 1" LJ >]o F'v･ AUXi for eaclL part･jtiori <>]o,£ i>

-3-




't DL,Iakot･o KiKucm I-Jt)1. 17

of E]', thcn r' FT A.

Fbr a classificat/iert A = <S, £ , t>, the theory

'1ih(A.)

g'enerated b.y A is definecl

by Th.(A) --

<E]A,FA> such that FA is the set of all sequents which are satisfied by

every, t,oken of A. It･ is sliewn ill l,:3] that Th(A) is a regula,r theory for aiiy A.

Definition 2.4. An irofbrTnat.io7i, co･nLell]t is a triple C == <A,T,N> of a classification

A = <S, >], t>, a regu]ar theory, 1- = <E].F>, and a subset AJ of S sueh that every

token in AJ' sat,isfies the constra,illts in Vi. T is called the theory of e, and an eleinellt

of ll･V

is called a norwLal token of e.

3. Modeloflnferences

An information context e = <A, T, N> represents t,hree kind of information about,

SA and £ A: focts tA, T"ales T, and a doTrea･in ofthe rules A:' , Oiir model of iiiferences

is modeled as revision of representations of part･ial knowledge about these information,

Definition 3.1. A represe･ntation of knou;ledge about e is a trip. Ie e' = <state', e'. A,''>

sucli that state' is a map froin SA tu Seq( £ A), V and Ar' are subset･s of Seq(ZA) aiid

SA, and stat,e'(a) is not ii! ir' for every a. (i N'

state', V, and N' represellts kiiowledge about P,=.4, eT, and Ar, and the repre-

sentation of the compleLe knowledge about e is <stateA,r･T,N>. state'(a) = (r,A>means a, tA "r arida FLA O' for every. 1' cT and ti c A. Tokens in N' are expected

to satisfy,J constraillts in V, and the last conditioii requires that our knowledge about,

tukens in iV" is coiisistent with F", CR,ule) in Peirce's terininelog.y is represented by

(I7', A> in V, and both (Case) and (Result) correspoiid to state'(a) for some a. c AJ''.

Iiiferences can be forniulat･ed as updating of representat ions of knowledge.

Definition 3.2. 1. Ded?tction is a res･'ision of state', Assume sta,t/e'(a) = <T,, A>,

g ¢ r. T' g r', and r' '-'

ye. 'Then.

we can infer a t g from these assumptions,

and state'(a) should be revised to <I] U {g},A>. This is deduction. Deduction is

an iiiference f'rom (Rule) ancl (Case) to CResult). 2. fnduct.･i,on cencerris e', Assume tliat r ti' A and seme tokeii in N' satisfies <1'.A>. Then, induction ls an ink)rencc of adding <I",A> t･o e'. <F,A> is a (Rule), which

is expect･ed to be satisfied by every element･ ef Ai',

3. Abd?.t,c'tion concerns stat･e' again. L･et･ a E iV' and st,ate'((z) = <F,A>. AssuTrie

that/ a t V, ye F' '-,/'. and g ¢ r. If a t p is true, w･e may deduce that a t Qil'i

by nieans of the censt,raint <{g},{･4i}). Abduc:tioTT is a revisioiL of sta,te'(a) te

<r u {p}, A>. 4. Des･ign is a reasoniTig about Art. Let･ T' be the theory <EA,I'>. iX speeificatien

ft}r desigll is described b.x.' a [l7'-consist,ent･ sequent <I", A>, which means an art･ifact

- ,1




A 1'Iatheinatical )'[ecle] of Deciuctive and Non-Deductive Inf'ereiices

FTa,ble

1 C,a±,egevy "f Infereiices

5

f'a.ct ritledo･main

deductivededuct･ion.' (1) (2)'n,on-ifeduc;tive･abductionir;ductionde.sign.

realizirig it, If "Je eanllot find a token "rhich realizes <F,A> in AI', "re have to

expand iV to A･''. T'his expa,nsion is design.

Example 3.3. Suppose that ai., ciL,, . . . , av. are bags of different kii]ds of beans, and

SB be the set ofbeans in a room. VLJ'e have a classification ofbeans B = <SB, XB, tB>such that lt)B = {cEt, c22, . . . , a,.} U {g} vLThere g means Tvhite, and t=B st･ands for the

relationship between SB aiid £ B. T'hat is, b FB p rep:esents b is white, and b tB ap-

means b belongs to a,j･. Iet TB = <EB,FB> be a t･heory such that FB consists of

<e,{ai,cE2,-･･,a{n}> a,nd <{a/i.a'J･},l> for i iL p'. rlr'hese

constraints meall that every

bean belon.ffs t,o one of the bags, and no bcan can bclong to two bags.

I.et A"' {.l SB be t/･he set of 1)eaiis aiid c}/-i. is the bag of white beans in Peirce;s

example. Then, iii n represenLaLion of knowledge e' = <st･at'.e', F', :V'>, (Rule) tCall

the

bcans fi'oni this bag a]"e white:' is represent･ed by a constraint <{ai}, {v}>, and (Case)[tthese

beans are from this bag'; and (R¢ sult) L`these

beans are white;' are represcnted

by propositions <{czi},e> <- stfitc,,'(b) a,nd <{-o},O> fi{ s(,ate'(b) for b E AJ'',, respective]y.

Deduction is an int'erence oi' addinsr ({c}/i}, e> to state'Cb), inductioii is aii activity of

addillg <{c}/i}, {g}> to e', and abductioii is a reasoning of adding <{g}, o> to state'Cb).'1]he

specific/at,icm being white is represented by a sequent <{g},X> alld desigiL is aii

act,ivit･y of expanding N' by adding a b c SB such that･ <{v･},, e> Sl state(b).

De,duct,ion and abduction are the san'ie kiiid ef inft]r'eiices irt t･]iu sellse tha･t thev v

a,re revision of sta,te'Ca), although the validity ef this iiifere,nce is guaraiJt･eed only ln

dedllc;tion. 71'hat is, they are illfereiices about focts. Induction is a revision o'f Y,xs'hich is croninletelv differeiit from abductioll in our niodel of infereiiees: induetiQn -v

is a inferenc:e a,bout r'rties C[].f. '1'able

1). Deductive infereiices about･ r'ules (1) alld a,

doma･in C2) wil] be discussed in the following, seetions.

However, there are lnterpretatiolls between induction, abduction, and desi.ff. n,

Assun'ie that we have a represelltation of knowledge about an inforina,tion context

e = (A, 1i, i'V> on which we do infereiice.

[["heorein 3.4. 1. IrLduct･i,on a･ri,d abci'tLcti,on a7'e inte･rpreta･ble mut'ua･"y.

9. Dc6'･ign ancl abducti,o･rt "v'(,' i,rzterTi･r'eta.blc mu,tually.

P･r'oof. '1 . Dell'ne a classiilcation utli by letting SA, = SA, £ A, be the set of sequenLs

on >]A, aild a tA, <r,A> if and oiily, if ct satisfies <r,A> in A. Let ei = <Ai,, Tl,AI>

be a,Ti inforsnation context for a t･heory 7ii. 7]'hen,, induction for Cthe representat･ionof kiiowledge about) e is abduction for (the represeiitaLion of knowledge abouL) Ci.

---･･ 5 -




6 )L･Iakoto ]<[Kuc:fTr Vol.17

For the converse, define a classiftcatien Ae., b,y S'A. = SA, EA, = £ U {dra : a C SA}

where cy. is a new symbol for a, E SA, and a tA. ･v if and only if g E £ and a g=.4 F,or gp ==. "/.. Lct e2 :T- <A2,7'2.Ar') be an infoi'mation cont ¢ xt. where V'2 is atheory

such that [-T,, is equal te F. [['he-, abductioll for C is iiiductioii for e2: a･ 1==A yo in e

correspollds to <{aa, .}, {9}> iU e2･ 2. Defiiie .ilL:3 by SA, = SA, £ A,, = E] J{pt･} for a llew s),,ml)ol u･, and a kA, vpif and enly if g c EA and a

if.tL ･p, or g =: lt. aiid a, c N. Let e3 = (A3,'Th,IXr> be

an informa,tion context t'or a theery, Tli. Then. design for e coT'responds to al)duction

fot' e3 since a E Ar iii e corresponds to a F==A., pL in C3. For the converse, consider an

inf()i'TTiaLioll c;ontext e4 == <A, 'TZ,

AJ'4> ",here 'Tli

is a new theory ancl N4 is thG set of

t,okens sa,t･isfv. ing a type ･vn in E]A. Thcn, a F=A vfi･ in e is represented by a F. IVIi in e.t.

([[]he last interpreta,tion is problerriat,ie sinc'e the {)i'igillal inforniatioii AJ' in e is lost

iii e4.) D

NVo can construct a div'ect interpretatioll bet"Teen iiiduc:tion and clesign siinilarly.

I'n our EorniulatioiL, whether an lnfcrence ts induction oi' abduction depeiids on the

choice of an illforinatlon coiite.xt. In [13], Peirce inentieriecl as t'ollo-,s.

It (= inductioi'i) has beeii seen that one and all are n'iere pr'ocesses for test-

ing hypet･hesis already in hand. The induct,ion adds nothing. .,. AbdueLioii.

oi.L t,he other haiid. is inerely pT'epai'at･or.v. It is the first step of scientilic

reasoning, as induction is the concluding step.

It follows t･ltat a hypot･hesis should be treated as both iuduct'ion and abductiorL.

A scientific law･ <T,A> is chosen as a hypothesis by abduct,ion ill e3,, and it will be

tested bv inductticm in C, tt

4. Regular Theories

In the defiiijt･ioii of a representatioii of knowledge e' = <si,at,c',t', Ai'> for an in-

formation coiitext･ C =- <A,, T, .N'>, we oniy assumed that state'(a,) is T'-c:oiisistcnL for

T' == (E:A,F">. Regularit},, tbr theories is a generaiizatien of the, basic conditions for

classical logic. "Jheii we assume t.hat theeries sliould satisfy regularit>., wc ca,n ex-

pand r' to soinet,hillg strortger autoTnatically. For exainple, b.y ineams of GIobal Cut,

we ca,n infer g F' p from g I-' 'e'1i a,nd 'e,)

V p. rlihis

is a kind of deductive iiifereiices

about rules. Similarly, Ai ' can be expanded, and this is a dcductive irLfere"ce about, a

doinain. In the rest of this paper, "re providct sornc rriat,heinatical inaehiiiery for t･he

expansiens ol; F" and A]'.

For theorias T aud 1'' with the sa,n'te t,ype set, we say 1' is equal te oT: weaker

than Tf alld denc]te T [ T' if every･ constraint･ of T is a, eoiistra,int of T'. biix a

classMcation AL := <S, E, t=>.

-- 6-




A Mathcmatieal ]L,iodel of Deductive and Non-Deductive lnferellces 7

Definition 4.1. 1. For a sub6et N of S, define a theory T(AI) cm E by I- ET("i) A

if and only. if <li,A> is satisfied by every to]cen ill A] . XX･:'e call TCAJ') the theory

oi' AJ' in A.,

2, For a theory. [I- on E], define a subsct Ai (7i) oF S as the set of tokeTis in S which

satisfy ever.y co. nstraint of T. "･Te call Ar(T) the set ef norma,l tokens of T in A.

'Lemma 4.2. 1;b･r a, subset Ai ofS a,nd a theory T on E,

i. T(N') is 'reyular',

2. T(AJ) I 7i(Apt) ･if Ari g AL

ge. T(g) is in,consisten.t,

4, T(S) --

Th(A),

5. N(T) [ IV(Tt) if T, ! Vi,

6'. !V(1-) = O if T is ･ineonsi.sc.te.･nt,

7. Ar(T) -S ifT-Th(A), 8, [li !

･T(AJ･ (l')).

9, N gl N(T(iV)).

Proposition 4.3. The foZlo7vivt,g are equi,va,lent:

1. <A, [l-, AJ'> i,s cLrt ･i'rL,fo･r7nation

context,

2. TI1i(AJ),

3. A[ g AJ･ (1").

For a inforrnation coiit･exL <A, 7-, Ai>, <A,T(A'), AT'> ai]d <A,T, AJ' (T)> are reii-

sions of information represented by e and AJ' . XAirJ'heii we get new informatioii about

consLra,ints and expaiid T to a stronger theory, we have 1,o i'educe A' i,o a smaller

seL. Oii the ceiitrary we have to reduce T to a, weaker theor'y when we get ncw

inforTtLation and expand N t,o a bigge･r set. [[his is a killd of non-monotonicit,",･r of

reasolling, and it is a geiieralizat･ion of t,he disc;ussions about c.'oTrd'm.onsensf: 'T'fta,soning

by Barwise and Seliginan in Lect,ure 19 of [31'/ and chcmyiTt,gr 'inj}/Jr'ination

co'rtte:'ts b,y

Barwise in Sect,ioii 4.3 oi' [1].

Definition 4.4. I. ]T'or' a set P of part,itions on E], define a theory. [l-(P) on Z by

r' FT(p) A if and only, if <I', A> ig <r',A'>, for any sequeiit <V',A'> c P.

2. For a theory T on >:･, deline a set P([l=) oE' partitio-s by <I',A> E P(7-) if and

only. if <r'',A'> fll <1i,A> for any sequeiit, <F',A'> siich tha,t li' tT A'.

Lemrna 4.5. Fbr anzt pa,rtition P ari,d theo･rz; T on £ ,

1. T(P) i,,s ･reguXaT',

2. 1･(P) [T<P') tf' P' [= P,

,9. 7-(O) ･is

'inconsz'stenL,

4. T(P) has no c'onstraint ･ij'

P 71s th,c set oj' atl .pa,rtit'ions,.

-7-

NII-Electronic Mbrary




8 tL,ta,koto KIKL.rcH[ Vo]. 17

5. P(T) ! P(T,) VTt [ T,

6, 1'(T) = szf Zf'T is incon,si,ste･nt, 7. P(T) is the set of aZl pa7'titions i.f

'T has

'no c;onst'r'ainL,

8. 1i!T(P(T)),

9. P -.T. P(T(P)).

It follows thai, a regular theory is conipletely determiiied by the set of c:onsist,e,nt

part･itions. The following are re]ations between tokens and partit,ic)ns.

Definition 4.6. 1. For a subset A] of S, let･ P(N) = {state(a) : a E N}.

2. For a, set P of partitions of X, let N(l') = {a E S : st･ate(a) E P}.

Lemma 4.7. fl)7' any sub,set A" ofS ancl parti,t'ion P of E],

1, P(N) ! P(Nt) if Ai !I Ar',

2, NCiJ)) ! N(Pt) ･Cf

P [ P,,

3, P(Ar(P)) g 1),

4, A] = Ar(p(Ar)).

The inutual relat･ionships betweell a theory, a set of normal tokeiis,, and a set ot'

partit･ions can be described as fo11ows.

Lemma 4.8. ebr angy ,stLbset N of S, partition P of E･, and th,eory T o･n E]･,

1. T(N) -T(P(A:･)),

2, T(P) :. T(AJ･ CP)), ,9. P(N) = P(T( A:･ )), 4. J'(A)(T))gP(T),

5. ,･V(T) = Ai(J)(7=)),.

6. A](P) =. AJ･(･7-(P)).

5. Regular Closures

'l'he

set, of t･heories and the set of subset,s of' S are ordered by [ and ! respec-

tive!y. "ie can prove the following proposition froiii leininas in t･he previous section,

Proposition 5.1. Th･e lmir of maps T and Ai is a. Galois connection between theo-

7'i･e.{ oTt, £ and subscts cije S, Tha,t i,s, T [ 7-(Ar) ･is cqzLi'vale'n,t to Ai (7-) g. Ai .joT' e'veT'y

theory T on £ and everzy subse[ A･' of S.

Since t]ie pair of' maps T and AJ' is a Galois connection, we can show that･ co]n-

positions of maps I' o A]' and N oT sat,isfies the conditions of closure operators. Since

T g T(N('T)) and N g N(T(A:')), To N and AJ' o7i can be regarded as supplemen-

tat,ien of information represented by T and N.

-- 8

NII-Electronic Mbrary



'


A Mathematical Model oi' Deductive and Non-I)educti'x,c lnferences 9

Definition 5.2. For any theory. T, the weakesL regular theory 1" such that 7' ! T'

is ealled the reguta･r closure of T and we clenote it by Reg(7-),

The exist･ence ofthe regular clusure is proved in [31, Reg(T) can be considered as

t･he set, of theor'eins provable from U', aiid ive regard a process of construct･ing R,eg(T)

from T as a sort･ of' deductive inference.

Definition 5.3. Let T be atheory on £ , VL['e sarv,r A is T-corrtplete ifeve'ry T-collsisteiit

sequent is realized by a t,oken in A.

The 7T-completeness of A means that A has sufiiciently maiiy tokens with re-

s,ard to T. X･N･"e shall prove tha,t T(Ai(I')) = Reg('T) for any theory V' on E: and any,

T-coinplete elassification A. Since G6del's completencss t,heorem can be expressed as

'[I]'h(ts[odel('1')) = [Iih(T) where Th(",Iodel(V')) is Lhe set of sentences which i's true in

every, model of T aiid Th(T) is the set oftheorems ofT, this claim can be corisidered

as an abstract versioii ef the ceinpleteness tlieoi'ein,

XKJe remark that･ T o P also sat,isttes the conditions of elosure operators, The

fo11owing leinma gives its eharacterization. '

Lemma 5.4. T(P(T)) = Reg(T. ) ,fbr a,ny theory '7i

on X.

Proof It･ is eiiough to show P(T) = .P(Reg(7-)) since P(T(P(7=))) = P([Z-) alld

T(P('T)) -- Reg(T) if a,nd only if .J'CT(P(1'))) - Reg(7]). It is clear that, P(RegCl')) g.P(1i) because T LI Reg(T). If P('1i) {l P(Reg(T)), we can obtain a regula,r theory

7i' such that 'T

! [t" and [t" is stric:Lly weaker than Reg(:r') which eontradiets the

cundition of Reg(7-). HerLce we havc J](T) gP(Reg(1i)). []

Now we can prove a t･heorem. which is a characterization of regular closures.

Theorem 5.5. Pbr a, 7'-con7,ptete class//fica･tton A, 7-(A) (T)) = Reg('T) for any thco･r:y

T on E.

Proo.tL Since A is 7--complete, J'(T) is the set of st･at,e descriptions of tokeris in N(1').rl]hen,

'1i(A"([l-))

=- =- T(P([IT)), hence we have T(A:' ('T)) = Reg(T). M

Barwisc and Seligman prov･ed in [3] that a theory T oii £ is regular if and onl,N.i

if t,herc ts a classification A such that T =:: Th(A), and tlicy call this the AbsLracL

Completeness Theorem. They proved their thcurein by (lefining A a,s the classifi-

c:at･ion Cla('7") = <Sci,{T),£ ci.<T), tci.{･T)> where Sc;i.(T) is the set･ of T- ¢ onsisLent

partitions, £ cl.q) = E, and <I"',A> F:ci.<T) cu if' a]'Ld cmly lf a E r. It is clanr t,haA,

Cla,('T) is T-complete, so our i,heorem is a genera,lizat･ion of the Ab6tract Cornpletelless

Theorem.

-9-




10 Makoto KIKUCHI Vol. 17

6. Conclusion

iLVe

disc'usrsed deductive and non-deductive infereiiees and proposed a rnat.heinat･-

ical model of inferences in which inferences are classified into three categories: ill-

forenccs about, facts, rules, and a clornain. XiVe show･ed also Lhcir inteT'pi'etability. aiid

mathematica! properties, and provcd a versioll of the eompleLeness theo:em whicli is

a c:hai"acter'iza,LioTi ef a sort of inforinat･ion revision.

Acknowledgments

[[he author would like to thallk Yuzuru Kakiida, Ic'hiro rslugasaka and Atsushi

Shin'iojinia fbr interesting discussions and coinments oii earlier ve,rsion of t･his paper.

[i]

!i2]

[3]

[4][5]

[6ii

[7]

[8]

[9]

[10]

[11]

[12][13]

[14]

[151

References

J. BuT:wise, IriformaLioti aiid IinpossibiliLies. Notre Dame J. of }r"ormal Logiic, 38, 1997,

488-515.

J. Barwise and J. Perr.y, S{btsation a･nd A"it?tde, (C･ainbridge, ]x,/[IZ[' Pt'e$s, 1983;

St,anford, CIST.I Pub]ications, 1999).

J. Barwisc and J. Selig'man J7'tformati･on Flo?v: The Logic of Distributed Systems,,(Cainbridge, Cambridge University Press,, 1997).

J. van Benthem, Reasonillg in Reverse, in [6], 2000, ix xi.

IT.I. Dretske, Kno?vledge an,d the 171o'tt,: of fn,fo･mna,tio･n, (Cambridge, "i'rlT Press, l981;

()･LSI 1'ub]icatioits, 1999).

P.A. Flach alld A.C. Kakas (Eds.), Abd'u.ctio'n a･nd fTtdtt,cti.onr Essays oTt thci.r' Jletatio･n

and fniesJ7'a･tion, CDordrecht. Klu",er Academic Press, 2000).P.A. Flach and A.C. Kakas, Abductive and lnductive Reasoning: Background and

lssues, in [6], 2000, 1-27.

A.CJ･. Kakas, R.A. Kowalski an(1 E Toiii, Abductive Logic: 1'rogramming', J. of LogLc

find CeTnptitatLon, 2, l992, 719-7'7C).

pt'I, I<ikuchi, Analysis and Design from a lvriewpoint of Inforn'iation Flow, in B. L6we,

NV. )L'Ialzkorii alld T. R;'lsch (Eds.), F?)'tt'rtdatto'n.s of FbTTn,a,l Scienc:es II, rl"s/ends in Lets.ic

17, (Dordrecht, Kluwer Acacleinic Publishers, 2003), 119-122.

S,H. }irugg!eton and L, De Ra,edt, Inducl,ii,･e Legic Prograinining: '1'lieory and Methods,

J, of Logic Programming, 19-20. 1994, 6Lt9-679.

C.S. I,'eirc/e, Deduct,ion, h]cluction, arid Hy, pot･hesis, Popular Scienee )･lonthly, 13, 1878,･e70-82. alsu in [1,1], 186-199.C･.S. Peirce, Of Reasoning in (Ileneral, in [15], ]S85, 11- 26.

C.S. Peirce, On the ]"ogic'. of Drawing HisLor/y rro]n Aiicient J')ocuinellts, Espe.cially firoiii"1'est,imonics,

i- [il5i/, 1901, 75-1]4.C.S. Peircc, The Essert･tial Peirce Vorume 1 (f867-189t9? (N,II. I{ouse]', C. Kloesel eds.),

(B]oorriington, lndiana U'nlversit･y Press, 1992).

C.S. Peirce, The tt]sse'n,t,iaZ l'e-z'rce VbiuTrLe 2 (IS99-191,9P (the 1'eirce Eclition Project,

m 10 ･-



the JapanAssociationforPhUosophy of Science

A Mathematical Model of Dedu ¢ tivc and Non-Dcductix,c !nferenees 11

[16]ed.), (Blooinington, I]dia,na University. Press. 1998).

H. Yosbikawa, General Desigii 'i'hcory as a Forinal Tlieor}, of Dc}sign,, in

and D. Gossard (Eds,). rnteli･igent aAP, l, (Amsterdam, NorLh-Holland,H.

Yt)shikawa

1989). 51-61.

11

all (Result) (Case)to (Result), (Case)and (Result) t,o (R.ule)

Documents

Transcript of all (Result) (Case)to (Result), (Case)and (Result) t,o (R.ule)